Introduction to Quantum Transport

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Introduction to Quantum Transport

Thomas Konstandin, IFAE Barcelona

June 23, 2009

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Outline

1 Motivation

The Basic Picture of EWBG

2 Semiclassical Transport

Thin wall / reflection picture Prerequisites and former approaches

Quantum-transport from Kadanoff-Baym equations

3 Models

EWBG in the MSSM EWBG in the nMSSM

4 Conclusions

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Sakharov conditions

Baryogenesis is one of the cornerstones of the Cosmological Standard Model and tries to explain the observed baryon asymmetry of the Universe (BAU)

η = nB − nB¯

s = 0.9 × 10⁻¹⁰.

The celebrated Sakharov conditions state the necessary ingredients for baryogenesis:

Sakharov conditions

Baryon number violation

Charge (C) and charge parity conjugation (CP) are no symmetries

non-equilibrium

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C- and B-violation: The sphaleron process

In the hot universe B- and C-violation is present due to sphaleron processes (’temperature induced instanton’).

The effective sphaleron vertex

Sphaleron bL bL tL sL sL cL

dL

uL νe

νµ ντ

HAll

∆B = 3, ∆L = 3, ∆NCS = 1 B− L conserving

B+ L violating

Exponentially suppressed after the phase transition (mW) Topological effect of the SU(2) gauge sector

EWPT is last chance of baryogenesis (φ < T ).

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First-order electroweak phase transition

Comments on bubble nucleation

Order parameter of the phase transition is the Higgs vev hhi During the phase transition the particles change their masses This is a violent process (v ∼ c) For EWBG, the PT has to be strong, φ & T

In the SM, there is only a cross-over (mHiggs > 60 GeV)

Kajantie, Laine, Rummukainen, Shaposhnikov (’96)

A first-order electroweak phase transition requires BSM physics.

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CP Violation in Mass Matrices

CP violation: Chargino masses in the MSSM and bMSSM

In the MSSM case the mass matrix of the charged higgsinos/winos is:

m=

M₂ g h₂(x^µ) g h₁(x^µ) µ_c

with M₂ and µc containing a CP-odd complex phase. The Higgs fields are during the phase transition space-time dependent.

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Picture of Electroweak Baryogenesis

Shaposhnikov (’87)

Higgs vev symmetric phase broken phase

lw

v_w

r hφi

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Picture of Electroweak Baryogenesis

CP-violating particle density (charginos, quarks, leptons) CP-violating effects

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Picture of Electroweak Baryogenesis

CP-violating particle density sphalerons active sphalerons not active

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Picture of Electroweak Baryogenesis

Cohen, Kaplan, Nelson (’90)

sphalerons active sphalerons not active

CP-violating particle density diffusion/transport

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Picture of Electroweak Baryogenesis

Cohen, Kaplan, Nelson (’90)

Higgs vev

CP-violating particle density CP violation & transport

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Outline

1 Motivation

3 Models

4 Conclusions

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Thin wall / reflection picture

Particle reflection, v = 0

Equilibrium with same temperatures is a steady-state solution.

Clasically, particles climb the wall if possible and just replace each other, distribution functions in the wall frame depend on

u_µp^µ= E . E,

q

p_z²− ∆m² E, pz

= =

m

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Particle reflection, v ≪ 1

For small velocities, a steady-state solution requires interactions (equilibration). The particle distribution functions in equilibrium depend an the wall frame on u_µp^µ= γ(E − vp_z).

E, q

p_z²− ∆m² E, p_z

= =

m

=

equilibration?, T changes?

equilibration?

shock front?

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Particle reflection, v ≈ 1

For large velocities, equilibration takes place behind the wall.

E, q

p_z²− ∆m² E, p_z

=

m exponential supressed

=

equilibration behind the wall rarefaction wave

Bodeker, Moore (’09)

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Particle reflection

A moving Higgs wall drives the plasma out of equilibrium in front of the wall due to reflections

behind the wall due to a change in the particle momenta The picture of particle reflection is only valid for thin walls, l_w ≪ 1/g²T.

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Prerequisites and former approaches

Outline

1 Motivation

3 Models

4 Conclusions

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Challenge in EWBG

Connection between macroscopic and microscopic scales In electroweak baryogenesis, one is rather interested in the case of thick wallswhich can be hardly treated in the pseudo-particle reflection picture due to multiple scatterings CP violation is a microscopic/quantumeffect produced by the interaction of single quanta in the plasma with the wall.

Transport is a macroscopic/classicaleffect based on statistical physics and particle densities.

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Classical Boltzmann Equations

The system is described by the particle distribution function n(x, v, t) on classical phase space.

The Boltzmann equations are based on a particle picture. If on the particles in the plasma acts an external force F , one obtains

n(x + vδt, v + Fδt, t + δt) − n(x, v, t) = collisions/interactions, and thus

(∂t+ v · ∇ + F · ∂v)n(x, v, t) = collisions/interactions.

How can CP violation be incorporated in this classical picture?

In which semi-classical limit can one obtain a phase-space from a quantum theory?

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Summary of Approaches to Transport

Semi-classical force / WKB approach

Joyce, Prokopec, Turok, hep-ph/9401352, hep-ph/9408339 Joyce, Cline, Kainulainen, hep-ph/9708393

Huber, Schmidt, hep-ph/0003122

Perturbative mixing / mass insertion approach

Carena, Moreno, Quiros, Seco, Wagner, hep-ph/0011055 Carena, Quiros, Seco, Wagner, hep-ph/0208043

Cirigliano, Profumo, Ramsey-Musolf, hep-ph/0603246

Kadanoff-Baym approach

Kainulainen, Prokopec, Schmidt, Weinstock, hep-ph/0105295 Konstandin, Prokopec, Schmidt, hep-ph/0410135

Konstandin, Prokopec, Schmidt, Seco, hep-ph/0505103 Huber, Konstandin, Prokopec, Schmidt, hep-ph/0606298

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Summary of former Approaches

Approach Cline/Joyce Carena/Moreno/Quiros Kainulainen Seco/Wagner CP-violation dispersion relation local source term

WKB perturbation theory

basis mass eigenbasis flavour eigenbasis quasi-particles charginos higgsinos/winos

transport classical classical

Boltzmann type diffusion mixing not included in the source

not in the diffusion

~ order second order first order comment momentum? basis?, finite?

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Outline

1 Motivation

3 Models

4 Conclusions

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Transport Theory and EWBG

Starting point of our formalism are the Kadanoff-Baym equations that are the statistical analogue to the Schwinger-Dyson equations:

Z

d⁴z S₀⁻¹(x_µ, z_µ) − Σ(x_µ, z_µ) S(zµ, y_µ) = ¹δ(x_µ− y_µ) where the self-energy Σ and the Green function S contain an additional 2 × 2 structure from the in-in-formalism

Σ = Σ^t Σ^>

Σ^< Σ^¯t

=Σ⁺⁺ Σ⁺⁻

Σ⁻⁺ Σ⁻⁻

, S = S^t S^>

S^< S^¯t

. Of these four entries only two are independent and S^< encodes the particle distribution function.

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Transport Theory and EWBG

In Wigner space we have

X_µ= (x_µ+ z_µ)/2, k_µ= FT (x_µ− z_µ).

Then the Kadanoff-Baym equations read e^{−i ♦}{S₀⁻¹− Σ, S} = 0.

with

2♦{A, B} := ∂X^µA∂kµB− ∂kµA∂X^µB

and all quantities are functions of X_µ and k_µ. The super-/sub- scripts <, >, R, A denote the additional 2 × 2 structure of the in-in formalism.

Without interaction

e^{−i ♦}{S₀⁻¹(z), S^<} = 0, e^{−i ♦}{S₀⁻¹(z), SA} = 0

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Particle densities

Using the KMS condition and the correct normalization one obtains in equilibrium

iS^<= 2π sign(k₀) δ(k²− m²) n(k₀), n(k₀) = 1 exp(βk₀) − 1 The particle density can (also away from equilibrium) be read off from S^< Z

k₀>0

dk₀

2π 2ik0S^<= n(k, t, x).

Thus the Wigner space yields in the semi-classical limit the usual phase space (however, not necessarily positive).

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A Simple Example

Free bosonic theory with one flavour and a constant mass:

e^{−i ♦}{S₀⁻¹, S^<} = 0, S₀⁻¹ = k²− m². The hermitian/anti-hermitian parts are the so called constraint/kinetic equations:

(k²− m²)S^<= 0, k^µ∂XµS^<= 0 which at low energies k^µ= (m, mv) is of Boltzmann type

m(∂t+ v · ∇)S^<= 0.

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Gradient Expansion

Consider

e^{−i ♦}{S₀⁻¹(z), S^<} = 0, S₀⁻¹ = k²− m²(z) Since the background in the MSSM is weakly varying

(l_w ≈ 20/T_c) the Moyal star product can be simplified by the semi-classical approximation

∂k∂X ≈ 1

T_cl_w ≈ 1

20 ≪ 1 → e^{−i ♦}≈ 1 − i♦ −1 2♦²· · · The simplest example for a transport equation in a varying background is for one bosonic flavour with real mass (up to first order in ♦)

(k²− m²)S^< = 0 (k^µ∂_µ−1

2(∂_zm²) ∂_k_z)S^< = 0.

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Fast walls and reflection

For fast walls, most particles in the plasma have high

p_z ≫ ∆m, 1/lw and hence the Kadanoff-Baym approach should agree with the reflection picture

Integration yields that the plasma depends behind the wall on u_µp^µ− uz

∆m² 2pz

+ O(∆m⁴/p_z⁴)

what for v ≈ 1 this agrees with the former result from reflections p_z →

q

p_z²− ∆m² ≈ pz− ∆m² 2pz

.

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CP violation

Prokopec, Schmidt, Weinstock (’01)

Consider a fermion with complex mass term S₀⁻¹ = k/− PLm(z) − PRm^∗(z) and

m(z) = |m(z)| e^iθ(z).

Then the kinetic equation up to second order in ♦ reads (s denotes the spin of the particle)

k^µ∂_µ−1

2(∂zm²) ∂kz − s

2k₀∂z(m²∂zθ) ∂kz

S_s^<(k^µ, z) = 0 which leads to CP-violating particle densities. This agrees with the findings of Cline/Joyce/Kainulainen in the WKB framework.

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Fermionic Systems

After spin projection the fermionic system of equations reads

“2i ˜k0−

k₀∂_t+ ~k_k· ∇_k

˜k₀

”

S₀^s−(2iskz+ s∂z) S₃^s−2im_he 2i

↼

∂z

⇀

∂kzS₁^s−2imae 2i

↼

∂z

⇀

∂kzS₂^s = 0

“

2i ˜k₀−k₀∂_t+ ~k_k· ∇_k

˜k0

”

S₁^s−(2skz−is∂_z) S₂^s−2imhe i 2

↼

∂z

⇀

∂kzS₀^s + 2mae i 2

↼

∂z

⇀

∂kzS₃^s = 0

“2i ˜k0−

k₀∂_t+ ~k_k· ∇_k

˜k₀

”

S₂^s+ (2skz−is∂z) S^s₁ −2m_he 2i

↼

∂z

⇀

∂kzS₃^s−2imae 2i

↼

∂z

⇀

∂kzS₀^s = 0

“

2i ˜k₀−k₀∂_t+ ~k_k· ∇_k

˜k₀

”

S₃^s−(2iskz+ s∂z) S₀^s + 2mhe i 2

↼

∂z

⇀

∂kzS₂^s −2mae i 2

↼

∂z

⇀

∂kzS₁^s = 0 ,

where S₀. . . S₃ are 2 × 2 matrices in flavour space and s denotes the spin.

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Transport in the Chargino sector

Konstandin, Prokopec, Schmidt (’04)

The chargino transport equations for the left/right handed densities are of the form

k^µ∂_µS^<+ i

2m², S^< + sources/forces = collisions Comments

S^< is a 2 × 2 flavor matrix

The term ₂ⁱm², S^< will lead to an oscillatory behaviour of the off-diagonal particle densities, similar to neutrino oscillations with frequency ∼ (m²₁− m₂²)/kz.

The source contains first order contributions that correspond to the sources in the approach of Carena et al.

The source contains second order contributions that correspond to the sources in the approach of Cline et al.

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Sources for EWBG

This approach resembles two mechanisms of EWBG from former approachesJoyce, Prokopec, Turok (’96) Cline, Joyce,

Kainulainen (’97,’00) Fromme, Huber (’06)

The dispersion shift source from the WKB approach:

S⁽²⁾∼n

m^†′′m− m^†m^′′, ∂_k_zS^<o .

Carena, Moreno, Quiros, Seco, Wagner (’00) Carena, Quiros, Seco, Wagner (’02)

Cirigliano, Profumo, Ramsey-Musolf (’06) Cirigliano, Ramsey-Musolf, Tulin, Lee (’06)

Sources from flavor mixing effects, e.g.

S⁽¹⁾ ∼h

m^†′m− m^†m^′, ∂kzS^<i .

CP violation from mixing appears only on the off-diagonal in the mass eigenbasis.

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Determination of the BAU

Huet, Nelson (’95)

The missing parts to determine the baryon asymmetry of the universe are:

h~

q~ q

Y and Sphaleron bL

bL tL sL sL cL

dL

uL νe

νµ ντ

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Diffusion and the Sphaleron

The originally used system of diffusion equations is of the form (for a recent treatment seeChung, Garbrecht, Tulin (’08))

v_wn^′_Q = D_qn^′′_Q− Γ_Y n_Q k_Q −n_T

k_T −n_H+ n_h k_H

− Γ_m n_Q k_Q −n_T

k_T

−6 Γss

2n_Q

kQ

−n_T kT

+ 9n_Q + nT

kB

+ Sources

· · ·

where n_Q, n_T, n_H, n_h denote particle densities, Γss, Γm, Γ_Y

interaction rates, kQ, kT, kH statistical factors and Dq a diffusion constant.

However, these equations are classical and back-reactions on the charginos cannot be taken into account in our approach.

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Advantages and Disadvantages

No ambiguities No divergences WKB and mixing effects

Flavor oscillations

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Advantages and Disadvantages

No ambiguities No divergences WKB and mixing effects

Flavor oscillations

No quantum transport in quark sector No quantum back-reactions on the charginos

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Outline

1 Motivation

3 Models

4 Conclusions

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EWBG in the MSSM

CP violation: Chargino masses in the MSSM

In the MSSM case the mass matrix of the charged higgsinos/winos is:

m=

M₂ g h₂(x^µ) g h₁(x^µ) µc

with M₂ and µc containing a CP-odd complex phase. The Higgs fields are during the phase transition space-time dependent.

Phase transition in the MSSM

In the MSSM the strength of the phase transition depends mostly on the loop effects of the bosons. A strong phase transition fulfilling the current mass bounds on the Higgs is possible if the stops are relatively light, mtop ∼ mstop.

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EWBG in the MSSM

Numerical Results

Konstandin, Prokopec, Schmidt, Seco (’05)

Parameters chosen: vw = 0.05, lw = 20/Tc, CP-phase maximal.

100 200 300 400

M 2 100

200 300 400

µc

mA=200

Due to the flavor oscillations, EWBG requires in the MSSM quasi-degenerate chargino masses.

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EWBG in the MSSM

Conclusions in the MSSM

CP violation in the MSSM is based on mixing between different flavors (charginos).

MSSM electroweak baryogenesis is a constrained scenario A light stop to acquire a strong first-order phase transition The condition µc ≈ M₂ .400 GeV of the a priori unrelated parameters M2 and µc

A large CP-violating phase that is testable by next generation EDM experiments

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EWBG in the nMSSM

Why is the nMSSM interesting?

Panagiotakopoulos, Pilaftsis (’02)

The nearly Minimal Supersymmetric Standard Model has the following effective superpotential

W_nMSSM = λˆS ˆH₁· ˆH₂−m₁₂²

λ Sˆ+ WMSSM, and has the virtues to solve the µ-problem of the MSSM by introducing a dynamical µ-term

µ = −λ hSi .

In this model singlet self-couplings are forbidden by a R’-symmetry.

The resulting model has neither problems with the stability of the hierarchy nor with domain walls (but λ might develop a Landau pole).

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EWBG in the nMSSM

CP violation: Chargino masses in the nMSSM

Huber, Konstandin, Prokopec, Schmidt (’06)

In the nMSSM the µ term contains a z-dependent complex phase µ(z) = −λ hSi = −λφs(z) e^iq^s^(z)

In the nMSSM second order sources dominate

The dynamical parameter µ = λ hSi leads to a dominating sources of WKB type

Charginos are generically non-degenerate (M2 &µ) Thin wall profiles

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EWBG in the nMSSM

Numerical Results

A numerical analysis of the BAU leads to the following result (sets passed LEP constraints and have a first order phase transition)

0 50 100 150 200 250 300 350

0 2 4 6 8 10

#

η10

# of models

0 50 100 150 200

0 2 4 6 8 10

#

η10

# of models

The left (right) plot shows the generated BAU for M₂ = 1 TeV (M₂ = 200 GeV). 50% (63%) of the models are in accordance with observation. The lower models fulfill the the EDM bounds with 1 TeV sfermion masses, 4.8 % (6.2 %).

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EWBG in the nMSSM

Conclusions in the nMSSM

CP violation in the nMSSM is based on mixing between different chiralities.

EWBG in the nMSSM is very promising

Strong first order phase transition due to tree-level dynamics η10&1 for most of the parameter space

EDMs eventually small due to small Arg (M2µc) two loop EDMs relatively small due to tan(β) ∼ O(1)

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Outline

1 Motivation

3 Models

4 Conclusions

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Final Remarks on EWBG

The Kadanoff-Baym equations provide a first principle approach to quantum transport.

They unite semi-sclassical force and mixing effects in one framework.

Electroweak baryogenesis is the main application of quantum transport equations so far.

Flavored leptogenesis?